Optimal. Leaf size=152 \[ -\frac {a}{12 d (a \sec (c+d x)+a)^{3/2}}+\frac {a}{2 d (1-\sec (c+d x)) (a \sec (c+d x)+a)^{3/2}}+\frac {7}{8 d \sqrt {a \sec (c+d x)+a}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a \sec (c+d x)+a}}{\sqrt {a}}\right )}{\sqrt {a} d}+\frac {9 \tanh ^{-1}\left (\frac {\sqrt {a \sec (c+d x)+a}}{\sqrt {2} \sqrt {a}}\right )}{8 \sqrt {2} \sqrt {a} d} \]
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Rubi [A] time = 0.14, antiderivative size = 152, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {3880, 103, 152, 156, 63, 207} \[ -\frac {a}{12 d (a \sec (c+d x)+a)^{3/2}}+\frac {a}{2 d (1-\sec (c+d x)) (a \sec (c+d x)+a)^{3/2}}+\frac {7}{8 d \sqrt {a \sec (c+d x)+a}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a \sec (c+d x)+a}}{\sqrt {a}}\right )}{\sqrt {a} d}+\frac {9 \tanh ^{-1}\left (\frac {\sqrt {a \sec (c+d x)+a}}{\sqrt {2} \sqrt {a}}\right )}{8 \sqrt {2} \sqrt {a} d} \]
Antiderivative was successfully verified.
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Rule 63
Rule 103
Rule 152
Rule 156
Rule 207
Rule 3880
Rubi steps
\begin {align*} \int \frac {\cot ^3(c+d x)}{\sqrt {a+a \sec (c+d x)}} \, dx &=\frac {a^4 \operatorname {Subst}\left (\int \frac {1}{x (-a+a x)^2 (a+a x)^{5/2}} \, dx,x,\sec (c+d x)\right )}{d}\\ &=\frac {a}{2 d (1-\sec (c+d x)) (a+a \sec (c+d x))^{3/2}}-\frac {a \operatorname {Subst}\left (\int \frac {2 a^2+\frac {5 a^2 x}{2}}{x (-a+a x) (a+a x)^{5/2}} \, dx,x,\sec (c+d x)\right )}{2 d}\\ &=-\frac {a}{12 d (a+a \sec (c+d x))^{3/2}}+\frac {a}{2 d (1-\sec (c+d x)) (a+a \sec (c+d x))^{3/2}}+\frac {\operatorname {Subst}\left (\int \frac {-6 a^4-\frac {3 a^4 x}{4}}{x (-a+a x) (a+a x)^{3/2}} \, dx,x,\sec (c+d x)\right )}{6 a^2 d}\\ &=-\frac {a}{12 d (a+a \sec (c+d x))^{3/2}}+\frac {a}{2 d (1-\sec (c+d x)) (a+a \sec (c+d x))^{3/2}}+\frac {7}{8 d \sqrt {a+a \sec (c+d x)}}-\frac {\operatorname {Subst}\left (\int \frac {6 a^6-\frac {21 a^6 x}{8}}{x (-a+a x) \sqrt {a+a x}} \, dx,x,\sec (c+d x)\right )}{6 a^5 d}\\ &=-\frac {a}{12 d (a+a \sec (c+d x))^{3/2}}+\frac {a}{2 d (1-\sec (c+d x)) (a+a \sec (c+d x))^{3/2}}+\frac {7}{8 d \sqrt {a+a \sec (c+d x)}}+\frac {\operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+a x}} \, dx,x,\sec (c+d x)\right )}{d}-\frac {(9 a) \operatorname {Subst}\left (\int \frac {1}{(-a+a x) \sqrt {a+a x}} \, dx,x,\sec (c+d x)\right )}{16 d}\\ &=-\frac {a}{12 d (a+a \sec (c+d x))^{3/2}}+\frac {a}{2 d (1-\sec (c+d x)) (a+a \sec (c+d x))^{3/2}}+\frac {7}{8 d \sqrt {a+a \sec (c+d x)}}-\frac {9 \operatorname {Subst}\left (\int \frac {1}{-2 a+x^2} \, dx,x,\sqrt {a+a \sec (c+d x)}\right )}{8 d}+\frac {2 \operatorname {Subst}\left (\int \frac {1}{-1+\frac {x^2}{a}} \, dx,x,\sqrt {a+a \sec (c+d x)}\right )}{a d}\\ &=-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+a \sec (c+d x)}}{\sqrt {a}}\right )}{\sqrt {a} d}+\frac {9 \tanh ^{-1}\left (\frac {\sqrt {a+a \sec (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{8 \sqrt {2} \sqrt {a} d}-\frac {a}{12 d (a+a \sec (c+d x))^{3/2}}+\frac {a}{2 d (1-\sec (c+d x)) (a+a \sec (c+d x))^{3/2}}+\frac {7}{8 d \sqrt {a+a \sec (c+d x)}}\\ \end {align*}
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Mathematica [C] time = 0.22, size = 90, normalized size = 0.59 \[ \frac {a \left (-9 (\sec (c+d x)-1) \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};\frac {1}{2} (\sec (c+d x)+1)\right )+8 (\sec (c+d x)-1) \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};\sec (c+d x)+1\right )-6\right )}{12 d (\sec (c+d x)-1) (a (\sec (c+d x)+1))^{3/2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.59, size = 546, normalized size = 3.59 \[ \left [\frac {27 \, \sqrt {2} {\left (\cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2} - \cos \left (d x + c\right ) - 1\right )} \sqrt {a} \log \left (\frac {2 \, \sqrt {2} \sqrt {a} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right ) + 3 \, a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right ) - 1}\right ) + 48 \, {\left (\cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2} - \cos \left (d x + c\right ) - 1\right )} \sqrt {a} \log \left (-8 \, a \cos \left (d x + c\right )^{2} + 4 \, {\left (2 \, \cos \left (d x + c\right )^{2} + \cos \left (d x + c\right )\right )} \sqrt {a} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} - 8 \, a \cos \left (d x + c\right ) - a\right ) + 4 \, {\left (31 \, \cos \left (d x + c\right )^{3} + 2 \, \cos \left (d x + c\right )^{2} - 21 \, \cos \left (d x + c\right )\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}}}{96 \, {\left (a d \cos \left (d x + c\right )^{3} + a d \cos \left (d x + c\right )^{2} - a d \cos \left (d x + c\right ) - a d\right )}}, -\frac {27 \, \sqrt {2} {\left (\cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2} - \cos \left (d x + c\right ) - 1\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {2} \sqrt {-a} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right )}{a \cos \left (d x + c\right ) + a}\right ) - 48 \, {\left (\cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2} - \cos \left (d x + c\right ) - 1\right )} \sqrt {-a} \arctan \left (\frac {2 \, \sqrt {-a} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right )}{2 \, a \cos \left (d x + c\right ) + a}\right ) - 2 \, {\left (31 \, \cos \left (d x + c\right )^{3} + 2 \, \cos \left (d x + c\right )^{2} - 21 \, \cos \left (d x + c\right )\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}}}{48 \, {\left (a d \cos \left (d x + c\right )^{3} + a d \cos \left (d x + c\right )^{2} - a d \cos \left (d x + c\right ) - a d\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 7.09, size = 229, normalized size = 1.51 \[ -\frac {\sqrt {2} {\left (\frac {48 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}}{2 \, \sqrt {-a}}\right )}{\sqrt {-a} \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )} - \frac {27 \, \arctan \left (\frac {\sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a} \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )} - \frac {3 \, \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}}{a \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right ) \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}} + \frac {2 \, {\left ({\left (-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a\right )}^{\frac {3}{2}} a^{4} + 12 \, \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a} a^{5}\right )}}{a^{6} \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}\right )}}{48 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 1.52, size = 504, normalized size = 3.32 \[ -\frac {\sqrt {\frac {a \left (1+\cos \left (d x +c \right )\right )}{\cos \left (d x +c \right )}}\, \left (1+\cos \left (d x +c \right )\right ) \left (-1+\cos \left (d x +c \right )\right )^{2} \left (48 \left (\cos ^{3}\left (d x +c \right )\right ) \sqrt {2}\, \sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \arctan \left (\frac {\sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {2}}{2}\right )+48 \sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \arctan \left (\frac {\sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {2}}{2}\right ) \sqrt {2}\, \left (\cos ^{2}\left (d x +c \right )\right )+27 \left (\cos ^{3}\left (d x +c \right )\right ) \sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \arctan \left (\frac {1}{\sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}}\right )-48 \cos \left (d x +c \right ) \sqrt {2}\, \sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \arctan \left (\frac {\sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {2}}{2}\right )+27 \sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \arctan \left (\frac {1}{\sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}}\right ) \left (\cos ^{2}\left (d x +c \right )\right )-48 \sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \arctan \left (\frac {\sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {2}}{2}\right ) \sqrt {2}+62 \left (\cos ^{3}\left (d x +c \right )\right )-27 \cos \left (d x +c \right ) \sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \arctan \left (\frac {1}{\sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}}\right )+4 \left (\cos ^{2}\left (d x +c \right )\right )-27 \sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \arctan \left (\frac {1}{\sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}}\right )-42 \cos \left (d x +c \right )\right )}{48 d \sin \left (d x +c \right )^{6} a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cot \left (d x + c\right )^{3}}{\sqrt {a \sec \left (d x + c\right ) + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\mathrm {cot}\left (c+d\,x\right )}^3}{\sqrt {a+\frac {a}{\cos \left (c+d\,x\right )}}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cot ^{3}{\left (c + d x \right )}}{\sqrt {a \left (\sec {\left (c + d x \right )} + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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